In \(3d\)  our proposal for the gauge theory on the \(k\) -wall is \(U(k)_{\frac{N-k}{2},\frac{N}{2}}^{\mathcal {N}=1}\)  with \(N\)  fundamentals and a quartic superpotential. Such proposal is a direct generalization of the ones in [1]}. One difference with respect to [1]} is the presence of "baryonic vacua", breaking the topological (or magnetic) \(U(1)\)  symmetry and hosting the NLSM \(S^1 \times Gr(J,F)\) , where \(J=0,\ldots ,k-1\)  and \(Gr(J,F)\)  is the Complex Grassmannian. Another difference with respect to [1]} is that for \(k>\frac{N}{2}\) , the semiclassical analysis of the mass deformed theory yields additional vacua. This is the same behaviour of \(Sp(N)\)  with \(F=N+2\)  flavors in [4]} and we expect that for \(k>\frac{N}{2}\)  the \(3d\)  quantum filed theory is strongly coupled and such additional vacua are not related to the phase transition relevant for the domain wall.The value \(k=\frac{N}{2}\) , for \(N\)  even, is special: we numerically find only two domain wall solutions instead of the expected \(\frac{N}{2}+1\)  vacua. A similar situation has been studied by [4]} when considering \(Sp(N)\)  with \(F=N+2\)  flavors domain walls, that have the same property of connecting vacua that lie on the real axis. In that situation, the seemingly single solution, was argued to be a superposition of different solutions. We believe that this is also the case for \(k=\frac{N}{2}\)  domain walls of \(SU(N)\)  with \(N\)  flavors. Moreover, if \(k=\frac{N}{2}\) , the \(3d\)  analysis is reliable and produces \(\frac{N}{2}+1\)  vacua, further supporting such expectation.
